Number of classes vs number of parameters/layers?
You are correct about your first assumption, but not about your second assumption.
More layers does not always mean better pattern detection. The analogy that in the deeper layers the network learns more complex features is an oversimplified one. It is true to some extent, although it is not enough to explain very complex architectures like GoogLeNet.
Moreover, using more layers than neccessary increases the risk of observing vanishing gradients. Because the network is too deep i.e. has too many layers, the gradient will be vanishingly small when backpropagating so many layers in the beginning will not get any significant update during training.